statistical forecasting models

8/12/2019 Statistical Forecasting Models

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Statistical Forecasting Models

(Lesson - 07)

Best Bet to See the Future


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Statistical Forecasting Models

Time Series Models : independent variableis time.

Moving Average Exponential Smoothening Holt-Winters Model

Explanatory Methods : independentvariable is one or more factor(s). Regression


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Time Series Models

Statistical Time Series Models are veryuseful for short range forecasting problemssuch as weekly sales.

Time series models assume that whateverforces have influenced the variables in

question (sales) in the recent past willcontinue into the near future.


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Time Series ComponentsA time series can be described by models based on the following

componentsT t Trend ComponentS t Seasonal Component

C t Cyclical ComponentI t Irregular Component

Using these components we can define a time series as the sum of itscomponents or an additive model

Alternatively, in other circumstances we might define a time series asthe product of its components or a multiplicative model oftenrepresented as a logarithmic model

t t t t t I C S T X

t t t t t I C S T X


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Components of Time Series Data

A linear trend is any long-term increase ordecrease in a time series in which the rate of

change is relatively constant. A seasonal component is a pattern that is

repeated throughout a time series and has arecurrence period of at most one year.

A cyclical component is a pattern within the timeseries that repeats itself throughout the time seriesand has a recurrence period of more than one year.


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Components of Time Series Data

The irregular (or random) component refers to changes in the time-series data that

are unpredictable and cannot be associatedwith the trend, seasonal, or cyclicalcomponents.


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Stationary Time Series Models

Time series with constant mean and varianceare called stationary time series.

When Trend, Seasonal, or Cyclical effects arenot significant then

a) Moving Average Models and b) Exponential Smoothing Modelsare useful over short time periods.


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Moving Average Models

Simple Moving Average forecast iscomputed as the average of the most recent

k-observations. Weighted Moving Average forecast is

computed as the weighted average of the

most recent k-observations where the mostrecent observation has the highest weight.


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Simple Moving Average Forecast

Weighted Moving Average Forecast k

Y ) Y ( E F

1 t

k t i i

t t

k

Y w ) Y ( E F

1 t

k t i i i

t t


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Weighted Moving Average

To determine bestweights and period(k) we can use

forecast accuracy. MSE = Mean

Square Error is agood measure forforecast accuracy.

RMSE = is thesquare root of theMSE.

Actual wMA(k=3)

Month Burglaries 100.00% =SUM(C4:C6) All weights should add-up exactly to 142 88 0.1 The further away from the forecast period43 44 0.3 weights: the lower is the weight44 60 0.6 Most recent observation has the highest weight45 56 58.0 =B5*$C$6+B4*$C$5+B3*$C$4

46 70 56.0 =B6*$C$6+B5*$C$5+B4*$C$447 91 64.8 =B7*$C$6+B6*$C$5+B5*$C$448 54 81.2 :49 60 66.7 :50 48 61.3 :51 35 52.2 :52 49 41.4 :53 44 44.7 :54 61 44.6 :

55 68 54.7 :56 82 63.5 :57 71 75.7 :58 50 74.059 59.5 Preliminary forecasted number of burglaries

MSE = 256.3 =SUMXMY2(B7:B20,C7:C20)/COUNT(B7:B20)RMSE = 16.01 =SQRT(C22)

Data: Evens - Burglar ies


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Weighted Moving Average Tools / Solver Set Target Cell: Cell containing RMSE value Equal to: Min By Changing Cells: Cells containing weights

Subject to constraints: Cell containing sum of the weight = 1 Options / (check) Assume Non-Negativity Solve ----- Keep Solver Solution ----- OK

Actual wMA(k =3)

Month Burglaries 100.00%

42 88 0.1

43 44 0.3

44 60 0.6

45 56 58.0

46 70 56.047 91 64.8

48 54 81.2

49 60 66.7

50 48 61.3

51 35 52.2

52 49 41.4

53 44 44.7

54 61 44.6

55 68 54.756 82 63.5

57 71 75.7

58 50 74.0

59 59.5

MSE = 256.3RMSE = 16.01


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Weighted Moving Average

Best weights for a given k (inthis case 3) is determined bysolver trough minimizingRMSE.

Same procedure could beapplied to models with differentk s and the one with lowestRMSE could be considered asthe model with best forecasting

period.

Actual wMA(k=3)

Month Burglaries 100.00%

42 88 0.0285 43 44 0.209344 60 0.7622 45 56 57.5

46 70 56.547 91 66.848 54 85.649 60 62.250 48 59.651 35 50.752 49 38.453 44 46.054 61 44.8

55 68 57.156 82 65.857 71 78.558 50 73.259 55.3

MSE = 250.6RMSE = 15.83


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Tools/ Data Analysis / Moving Average Input Range: Observations with title (No time) Output Range: Select next column to the input

range and 1-Row below of the first observation Chart misaligns the forecasted values!

Forecasted 59th month is aligned with 58th month

Months Crime k = 3 errors

50 48

51 35 #N/A #N/A

52 49 #N/A #N/A

53 44 44.00 #N/A

54 61 42.67 #N/A

55 68 51.33 6.33

56 82 57.67 8.21

57 71 70.33 10.5958 50 73.67 9.13

59 67.67 12.32

Moving Average

010203040

5060708090

50 51 52 53 54 55 56 57 58

Months

C r i m e s

Actual

Forecast


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Exponential Smoothing

Exponential smoothing is a time-series smoothingand forecasting technique that produces an

exponentially weighted moving average in whicheach smoothing calculation or forecast is dependentupon all previously observed values.

The smoothing factor is a value between 0and 1, where closer to 1 means more weigh to therecent observations and hence more rapidlychanging forecast.


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Exponential Smoothing Model

where:F t= Forecast value for period t

Y t-1 = Actual value for period t-1Ft-1 = Forecast value for period t-1

= Alpha (smoothing constant)

) F Y ( F F 1 t 1 t 1 t t

1 t 1 t t F ) 1 ( Y F or


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Tools/ Data Analysis / ExponentialSmoothing.

Input Range: Observations with title (Notime)

Output Range: Select next column to theinput range and first Row of the firstobservation

Damping Factor: 1- (not )

Month Crimes alpha=0.7

50 48 #N/A

51 35 48.0

52 49 38.9

53 44 46.0

54 61 44.6

55 68 56.1

56 82 64.4

57 71 76.7

58 50 72.7

59 ? 56.8

Exponential Smoothing

0

10

20

30

40

50

60

70

80

90

50 51 52 53 54 55 56 57 58 59

Months

C r i m e s

Actual

Forecast


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To determine best we canuse forecastaccuracy.

MSE = MeanSquare Error is agood measure for

forecastaccuracy.

A B C D

1 Month Crime 0.7

2 50 48 #N/A

3 51 35 48.00 ! Actual observation B2

4 52 49 38.905 53 44 45.97

6 54 61 44.59

7 55 68 56.088 56 82 64.429 57 71 76.73

10 58 50 72.7211 59 ? 56.82 =$C$1*B10+(1-$C$1)*C1012

13 MSE = 193.0 =SUMXMY2(B3:B10,C3:C10)/COUNT(B3:B10)


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Holt-Winters Model

The Holt-Winters forecasting model could be used in forecasting trends. Holt-Winters

model consists of both an exponentiallysmoothing component (E, w) and a trendcomponent (T, v) with two differentsmoothing factors.


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Holt-Winters Model

where:F t+k = Forecast value k periods from tY t-1 = Actual value for period t-1E t-1 = Estimated value for period t-1T t = Trend for period tw = Smoothing constant for estimatesv = Smoothing factor for trend

k = number of periods

) T E )( w 1 ( wY E 1 t 1 t 1 t t

1 t 1 t t t T ) v 1 ( ) E E ( v T

t t k t kT E F

1. E 1 and T 1 arenot defined.

2. E 2 = Y 2 3. T 2 = Y 2 Y1


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Holt-Winters Model

E_2 = Y_2 and T_2 = (Y_2-Y_1) E_12 = $D$1*C14+(1-$D$1)*(D13+E13) T_12 = $E$1*(D14-D13)+(1-$E$1)*E13

F_13 = D14+E14

A B C D E1 w = 0.7 0.5 = v2 Month Sales E T F3 1 4.8 N/A N/A4 2 4.0 4.0 -0.85 3 5.5 4.8 0.0 3.26 4 15.6 12.4 3.8 4.87 5 23.1 21.0 6.2 16.18 6 23.3 24.5 4.8 27.29 7 31.4 30.8 5.6 29.3

10 8 46.0 43.1 8.9 36.311 9 46.1 47.9 6.9 52.112 10 41.9 45.8 2.4 54.8

13 11 45.5 46.3 1.4 48.114 12 53.5 51.8 3.5 47.715 13 55.24

Holt-Winter Forecasting

0.010.020.030.040.050.060.0

1 2 3 4 5 6 7 8 9 1 0 1 1 1 2 1 3

Months

S a l e s

SalesF


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Holt-Winters Model

Set E (smoothing component), T (trendcomponent), and F (forecasted values) columns

next to Y (actual observations) in the samesequence Determine initial w and v values Leave E,T &F blanc for the base period (t=1) Set E 2 = Y 2 Set T 2 = Y 2-Y 1 Note: (F 2 is blanc )


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Holt-Winters Model

Formulate E 3 = w*Y 3 + (1-w)*(E 2+T 2) Formulate T 3 = v*(E 3-E2) + (1-v)*T 2 Formulate F 3 = E 2 + T 2 Copy the formulas down until reaching one

cell further than the last observation (Y n). Compute MSE using Y s and F s Use solver to determine optimal w and v.


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Holt-Winters Model

Solver set up for Holt Winters: Target Cell : MSE (min) Changing Cells : w and v Constrains : w = 0v = 0


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Forecasting with Crystal Ball

CBTools / CB Predictor [Input Data] Select

Range, First Raw, First Column Next [Data Attribute] Data is in Next [Method Gallery] Select All Next

[Results] Number of periods to forecast [ 1]Select Past Forecasts at cell Run

periods, etc.


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Forecasting with Crystal Ball Year Actual Revenue

1975 5.0 Actual Revenues of EASTMAN KODAC1976 5.4 Data: EASTMANK1977 6.01978 7.01979 8.01980 9.7

1981 10.31982 10.81983 10.21984 10.61985 10.61986 11.51987 13.31988 17.01989 18.41990 18.9

1991 19.41992 20.21993 16.31994 13.71995 15.31996 16.21997 14.51998 13.41999 14.1


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Forecasting with Crystal Ball

Forecast:

Date Lower:

5% Forecast Upper: 95%

2000 11.9 14.4 17.0

MethodErrors:

Method RMSE MAD MAPE

Best:

DoubleExponentialSmoothing 1.5043 0.9871 7.68%

2nd: Single Exponential

Smoothing 1.5147 1.1566 9.03%

3rd: Single Moving

Average 1.5453 1.2042 9.40%

4th: Double Moving

Average 2.0855 1.592 11.16%

Method Parameters:

Method Parameter Value

Best:

Double ExponentialSmoothing Alpha 0.999

Beta 0.051

2nd: Single Exponential

Smoothing Alpha 0.999

3rd: Single Moving Average Periods 1

4th: Double Moving Average Periods 2

Actual Revenue

0.0

5.0

10.0

15.0

20.0

25.0

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1 9 8 1

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1 9 9 1

1 9 9 3

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1 9 9 9

Data

Fitted

Forecast

Upper: 95%

Lower: 5%

StudentEdition

StudentEdition


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Performance of a Model

Performance of a model is measured by Theils U .

The Theil's U statistic falls between 0 and 1.When U = 0, that means that the predictive

performance of the model is excellant and

when U = 1 then it means that the forecasting performance is not better than just using thelast actual observation as a forecast.


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Theils U versus RMSE

The difference between RMSE (or MAD orMAPE) and Theils U is that the formars aremeasure of fit ; measuring how well modelfits to the historical data.The Theil's U on the other hand measureshow well the model predicts against a naivemodel. A forecast in a naive model is done byrepeating the most recent value of the variableas the next forecasted value.


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Choosing Forecasting Model

The forecasting model should be the one withlowest Theil s U.

If the best Theil s U model is not the same asthe best RMSE model then you need to runCB again by checking only the best Theil s U

model to obtain forecasted value.P.S. CB uses forecasting value of the lowestRMSE model (best model according CB)!


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Determining Performance

Theils U determins the forecasting performance of the model.

The interpretation in daily language is asfollows:Interpret (1- Theil U) 1.00 0.80 High (strong) forecasting power0.80 0.60 Moderately high forecasting power0.60 0.40 Moderate forecasting power0.40 0.20 Weak forecasting power0.20 0.00 Very weak forecasting power


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Regression or Time Series Forecast

Here is the guiding principle when to applyRegression and when to apply Time Series Forecast.

As some thing changes (one or more independentvariables) how does another thing (dependentvariable) change is an issue of directional relationshipFor directional relationships we can use regression.

If the independent variable is TIME (as time changeshow does a variable change) Then we can use eitherregression or time series forecasting models


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Explanatory Methods

Simple Linear Regression Model : Thesimplest inferential forecasting model is the

simple linear regression model, where time(t) is the independent variable and the leastsquare line is used to forecast the futurevalues of Y t.


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Regression in Forecasting Trends

where:Y t = Value of trend at time t

0 = Intercept of the trend line

1 = Slope of the trend linet = Time (t = 1, 2, . . . )

t 1 0 t t t ) Y ( E F


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Regression in Forecasting

Seasonality Many time series have distinct seasonal pattern. ( For

example room sales are usually highest around summer periods.)

Multiple regression models can be used to forecast a timeseries with seasonal components.

The use of dummy variables for seasonality is common. Dummy variables needed = total number of seasonality 1

For example: Quarterly Seasonal: 3 Dummies are needed, MonthlySeasonal: 11 Dummies needed, etc. The load of each seasonal variable (dummy) is compared to the

one which is hidden in intercept.


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Regression in Forecasting

Seasonalityt 3 4 2 3 1 2 1 0 t t Q Q Q t ) Y ( E F

where:Q1 = 1 , if quarter is 1, = 0 otherwiseQ2 = 1 , if quarter is 2, = 0 otherwise

Q3 = 1 , if quarter is 3, = 0 otherwise2 = the load of Q 1 above Q 4

0 = the overall intercept + the load of Q 4 t = Time (t = 1, 2, . . . )


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Seasonal RegressionMegaWattsPower Loa Year Q1 Q2 Q3

106.8 1973.1 1 0 089.2 1973.2 0 2 0

110.7 1973.3 0 0 391.7 1973.4 0 0 0

108.6 1974.1 1 0 098.9 1974.2 0 2 0

120.1 1974.3 0 0 3102.1 1974.4 0 0 0113.1 1975.1 1 0 0

94.2 1975.2 0 2 0120.5 1975.3 0 0 3107.4 1975.4 0 0 0116.2 1976.1 1 0 0104.4 1976.2 0 2 0131.7 1976.3 0 0 3117.9 1976.4 0 0 0

Seasonal Regression

80.0085.00

90.0095.00

100.00105.00110.00115.00120.00125.00130.00135.00

1 9 7 3

. 1

1 9 7 3

. 2

1 9 7 3

. 3

1 9 7 3

. 4

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. 1

1 9 7 4

. 2

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Year/Quarter

P o w e r

Predicted Power

Load

Actual Power

Load

E(Y_Q1) = -10801.6 + 5.52 * Year.1 + 8.06E(Y_Q2) = -10801.6 + 5.52 * Year.2 + -3.50

E(Y_Q3) = -10801.6 + 5.52 * Year.3 + 5.51

E(Y_Q4) = -10801.6 + 5.52 * Year.4


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Introduction to Optimization

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